Simplify; express your answer in exponential form. Assume $z\neq 0, a\neq 0$. $\dfrac{{(z^{-1}a^{3})^{5}}}{{(z^{4}a^{4})^{5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-1}a^{3})^{5} = (z^{-1})^{5}(a^{3})^{5}}$ On the left, we have ${z^{-1}}$ to the exponent ${5}$ . Now ${-1 \times 5 = -5}$ , so ${(z^{-1})^{5} = z^{-5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-1}a^{3})^{5}}}{{(z^{4}a^{4})^{5}}} = \dfrac{{z^{-5}a^{15}}}{{z^{20}a^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-5}a^{15}}}{{z^{20}a^{20}}} = \dfrac{{z^{-5}}}{{z^{20}}} \cdot \dfrac{{a^{15}}}{{a^{20}}} = z^{{-5} - {20}} \cdot a^{{15} - {20}} = z^{-25}a^{-5}$